Modern wind turbines are commonly used to supply electricity into the electrical grid. Wind turbines of this kind generally comprise a rotor with a rotor hub and a plurality of blades. The rotor is set into rotation under the influence of the wind on the blades. The rotation of the rotor shaft either directly drives the generator rotor (“directly driven”) or through the use of a gearbox.
A variable speed wind turbine may typically be controlled by varying the generator torque and the pitch angle of the blades. As a result, aerodynamic torque, rotor speed and electrical power will vary.
A common prior art control strategy of a variable speed wind turbine is described with reference to FIG. 1. In FIG. 1, the operation of a typical variable speed wind turbine is illustrated in terms of the pitch angle (β), the electrical power generated (P), the generator torque (M) and the rotational velocity of the rotor (ω), as a function of the wind speed.
In a first operational range, from the cut-in wind speed to a first wind speed (e.g. approximately 5 or 6 m/s), the rotor may be controlled to rotate at a substantially constant speed that is just high enough to be able to accurately control it. The cut-in wind speed may be e.g. approximately 3 m/s.
In a second operational range, from the first wind speed (e.g. approximately 5 or 6 m/s) to a second wind speed (e.g. approximately 8.5 m/s), the objective is generally to maximize power output while maintaining the pitch angle of the blades constant so as to capture maximum energy. In order to achieve this objective, the generator torque and rotor speed may be varied so as keep the tip speed ratio λ (tangential velocity of the tip of the rotor blades divided by the prevailing wind speed) constant so as to maximize the power coefficient Cp.
In order to maximize power output and keep Cp constant at its maximum value, the rotor torque may be set in accordance with the following equation:T=k·ω2, whereink is a constant, and ω is the rotational speed of the generator. In a direct drive wind turbine, the generator speed substantially equals the rotor speed. In a wind turbine comprising a gearbox, normally, a substantially constant ratio exists between the rotor speed and the generator speed.
In a third operational range, which starts at reaching nominal rotor rotational speed and extends until reaching nominal power, the rotor speed is kept constant, and the generator torque is varied to such effect. In terms of wind speeds, this third operational range extends substantially from the second wind speed to the nominal wind speed e.g. from approximately 8.5 m/s to approximately 11 m/s.
In a fourth operational range, which may extend from the nominal wind speed to the cut-out wind speed (for example from approximately 11 m/s to 25 m/s), the blades may be rotated (“pitched”) to maintain the aerodynamic torque delivered by the rotor substantially constant. In practice, the pitch is actuated such as to maintain the rotor speed substantially constant. At the cut-out wind speed, the wind turbine's operation is interrupted.
In the first, second and third operational ranges, i.e. at wind speeds below the nominal wind speed (the sub-nominal zone of operation), the blades are normally kept in a constant pitch position, namely the “below rated pitch position”. Said default pitch position may generally be close to a 0° pitch angle. The exact pitch angle in “below rated” conditions however depends on the complete design of the wind turbine.
The before described operation may be translated into a so-called power curve, such as the one shown in FIG. 1. Such a power curve may reflect the optimum operation of the wind turbine under steady-state conditions. However, in non-steady state (transient) conditions, the operation may not necessarily be optimum.
As further background, basic aerodynamic behaviour of (the blades of) a wind turbine is explained with reference to FIGS. 2a-2e. 
In FIG. 2a, a profile of a wind turbine blade is depicted in operation. The forces generated by the aerodynamic profile are determined by the wind that the profile “experiences”, the effective wind speed Ve. The effective wind speed is composed of the axial free stream wind speed Va and the tangential speed of the profile Vt. The tangential speed of the profile Vt is determined by the instantaneous rotor speed ω and the distance to the centre of rotation of the profile, the local radius r, i.e. Vt=ω·r.
The axial free stream wind speed Va is directly dependent on the wind speed Vw, and on the speed of the wind downstream from the rotor Vdown, that is Va=½(Vw+Vdown). The axial free stream wind speed may e.g. be equal to approximately two thirds of the wind speed Vw.
The resultant wind flow, or effective wind speed Ve, generates lift L and drag D on the blade. A blade may theoretically be divided in an infinite number of blade sections, each blade section having its own local radius and its own local aerodynamic profile. For any given rotor speed, the tangential speed of each blade section will depend on its distance to the rotational axis of the hub (herein referred to as local radius).
The lift generated by a blade (section) depends on the effective wind speed Ve, and on the angle of attack of the blade (section) a, in accordance with the following formula:
      L    =                  1        2            ⁢              ρ        ·                  C          L                    ⁢                        V          e          2                ·        S              ,whereinρ is the air density, Ve is the effective wind speed, CL is the lift coefficient (dependent on the angle of attack α) and S is the surface of the blade section.
Similarly, the drag D generated by a blade section can be determined in accordance with the following equation:
      D    =                  1        2            ⁢              ρ        ·                  C          D                    ⁢                        V          e          2                ·        S              ,wherein CD is the drag coefficient, dependent on the angle of attack α.
For an entire wind turbine blade, the contribution to lift and drag of each blade section may be summed to arrive at the total drag and lift generated by the blade.
Both the drag coefficient CD and the lift coefficient CL depend on the profile or the blade section and vary as a function of the angle of attack of the blade section. The angle of attack α may be defined as the angle between the chord line of a profile (or blade section) and the vector of the effective wind flow.
FIG. 2b illustrates in a very general manner how the lift coefficient and drag coefficient may vary as a function of the angle of attack of a blade section. Generally, the lift coefficient (reference sign 21) increases to a certain maximum at a so-called critical angle of attack 23. This critical angle of attack is also sometimes referred to as stall angle. The drag coefficient (reference sign 22) may generally be quite low and starts increasing in an important manner close to the critical angle of attack 23. This rapid change in aerodynamic behaviour of a profile or blade section is linked generally to the phenomenon that the aerodynamic flow around the profile (or blade section) is not able to follow the aerodynamic contour and the flow separates from the profile. The separation causes a wake of turbulent flow, which reduces the lift of a profile and increases the drag significantly.
The exact curves of the lift coefficient and drag coefficient may vary significantly in accordance with the aerodynamic profile chosen. However, in general, regardless of the aerodynamic profile chosen, a trend to increasing lift up until a critical angle of attack and also a rapid increase in drag after a critical angle of attack can be found.
In accordance with FIG. 2a, the tangential force generated by a blade section is given by T=L·sin(α+ϑ)−D·cos(α+ϑ), wherein ϑ is the pitch angle and α is the angle of attack. The pitch angle may be defined as the angle between the rotor plane and the chord line of a profile. Integrating the tangential force distribution over the radius provides the driving torque.
In order to increase the torque generated by the rotor, the angle of attack of any blade section is preferably kept below the critical angle of attack such that lift may be higher and drag may be lower.
It should be borne in mind that the angle of attack of each blade section depends on the tangential speed of the specific rotor blade section, the wind speed, the pitch angle and the local twist angle of the blade section. The local twist angle of a blade section may generally be considered constant, unless some kind of deformable blade is used. The tangential speed of the rotor blade section depends on the rotor speed (angular velocity of the rotor which is obviously the same for the whole blade and thus for each blade section) and on the distance of the blade section to the rotational axis.
For a given pitch angle, it follows that the angle of attack is determined by the tip speed ratio:
  λ  =                    ω        ·        R                    V        w              .  From this, it follows that the torque generated by a rotor blade section may become a rather complicated function of the instantaneous tip speed ratio and the pitch angle of the blade.
This complicated relationship between the tip speed ratio, pitch angle, and performance (e.g. torque) of the rotor may be depicted in a three-dimensional figure, such as the one shown as FIG. 2c. 
For every rotor blade section, the torque generated may be depicted by a line such as one of the lines shown in FIG. 2d of constant pitch angle or similar. (This FIG. 2d is from Erich Hau, “Wind Turbines, Fundamentals, Technologies, Application, Economics”.)
These lines depict the torque coefficient (CQ), i.e, the ratio between the torque developed by the wind turbine rotor and the maximum theoretical torque. Such lines may be obtained by a cross-section of the plane of constant pitch angle with the plane depicted in FIG. 2c. For each pitch angle, there is a certain critical tip speed ratio for which maximum torque generation is achieved.
This may be illustrated in an alternative manner, such as shown in FIG. 2e. For a given tip speed ratio, e.g., λ1, there is a certain critical pitch angle ϑcrit. This critical pitch angle gives the before-mentioned critical angle of attack for the given tip speed ratio. Below the pitch angle, stall may occur. At the same time, at the critical pitch angle, the generated torque is maximum.
During start-up of a wind turbine from standstill to normal operation, the blades generally have to be rotated from a feather position towards an operational position. With reference to FIG. 3a, the blade may be rotated from a feathered position 32 towards an operational position 31, which in this case is shown to be an operation with zero degrees pitch, i.e. the chord line of the blade section lies in the plane of rotation.
However, the final operational position of the blade may depend upon the wind conditions at that specific moment. Start-up may be performed under conditions of generally low wind, e.g. because the wind turbine operation was interrupted because of relatively low wind speeds. Start-up may also be performed under conditions of relatively high wind speeds, e.g. because the wind turbine operation was interrupted at high wind speeds in order to avoid high loads on the wind turbine. Start-up may also be formed in any intermediate wind situation. This may be the case when wind turbine operation was interrupted for e.g. maintenance.
As the blade is rotated from a feathered position towards an operational position, the pitch angle will vary between approximately 90° (feather position) and the final pitch angle in operation, which could generally be in the range of e.g. 0°-30°. During start-up, the speed of rotation and thus the tangential velocity of a blade section may be relatively small as compared to normal operation. And as mentioned before, the wind speed could vary from very low to very high. The angle of attack of the blades may thus also vary widely.
During start-up, as the wind turbine rotor starts rotating and increases the rotor speed, and as the blade is pitched, the angle of attack of the blade undergoes a complicated transition. In this sense, the angle of attack of the blade may be defined as the angle of attack of a representative blade section. As explained before, each blade section may have its own angle of attack at any given moment. The representative blade section may be chosen at e.g. 25% of the blade length.
In a known method of starting-up a wind turbine, the blade is generally rotated at a given constant rate, regardless of the prevailing wind conditions, until the rotor starts rotating. This pitch rate is generally chosen to be relatively low, in order to avoid stall when the wind turbine is started up in conditions of relatively high wind speeds.
When the rotor starts rotating, the pitch rate is changed to a predetermined, higher second level, regardless of the prevailing wind conditions. This pitch rate is chosen to be higher so that the blades can speed up faster. If however, the pitch rate were chosen too high, under low wind speed conditions, stall could occur.
In the final phase of start-up, a look-up table of pitch angles vs. generator speed (or rotor speed) may be used until proper operation is started, i.e. until the generator is generating electrical power.
Using this method, start-up may take longer than needed, which means that less electrical power is generated.
In another method, disclosed in U.S. Pat. No. 4,160,170, both wind speed and rotor speed are taken into account during start-up. For different rotor speeds, blade pitch angles are established depending on the wind speed.
There still exists a need for a method of starting up a wind turbine which effectively reduces start-up time while simultaneously keeping blade loads etc. under control e.g. by avoiding stall of the wind turbine blade. This need is fulfilled in various examples of the present disclosure.